How to Calculate Poisson Distribution Probabilities?

In summary, the Poisson Distribution is a discrete probability distribution used to model the number of independent events occurring within a specified time interval or area, with a known average rate of occurrence. It is characterized by being discrete and unbounded on the positive side, and defined by a single parameter, lambda. It differs from other distributions in its discreteness, single parameter, and use for rare events. Real-world applications include insurance, biology, finance, and traffic analysis. To solve problems using the Poisson Distribution, one must know lambda to use the Poisson probability formula or statistical software/tables.
  • #1
mathsforumuser
1
0
Hi guys I got a question on the poisson distribution and never previously done stats at all.

It follows:

The mean count of a radioactive substance is 25 disintegrations per minute. Using the Poisson distribution, find the probability that, in a time of 12 seconds, there are-

i) No disintegrations

ii) More than 2 disintegrations
 
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  • #2
Hi mathsforumuser,

Welcome to MHB! :)

When working with any discrete probability distribution, we often need to use the probability mass function. For a Poisson distribution this is:

\(\displaystyle P[X=k]=\frac{\lambda^k e^{-\lambda}}{k!}\)

For this problem, what is $\lambda$? What is $k$?
 

FAQ: How to Calculate Poisson Distribution Probabilities?

What is the Poisson Distribution?

The Poisson Distribution is a discrete probability distribution that is used to model the number of events that occur within a specified time interval or within a certain area, when the events are independent and the average rate of occurrence is known.

What are the key characteristics of the Poisson Distribution?

The key characteristics of the Poisson Distribution are that it is a discrete distribution, meaning it only takes on integer values, and it is unbounded on the positive side, meaning it can take on values of infinity. It is also defined by a single parameter, lambda, which represents the average rate of occurrence of the event.

How is the Poisson Distribution different from other probability distributions?

The Poisson Distribution is different from other probability distributions, such as the Normal Distribution, because it is a discrete distribution, while others are continuous. It also has only one parameter, lambda, while others may have multiple parameters. Additionally, the Poisson Distribution is used to model the occurrence of rare events, while other distributions may be used for more common events.

What are some real-world applications of the Poisson Distribution?

The Poisson Distribution has many real-world applications, such as in insurance and risk management to calculate the likelihood of rare events, in biology to model the occurrence of mutations or diseases, and in finance to model the arrival of stock market orders. It can also be used in traffic analysis to predict the number of cars arriving at a certain location in a given time period.

How do I use the Poisson Distribution to solve problems?

To use the Poisson Distribution to solve problems, you will need to know the average rate of occurrence of the event, lambda. From there, you can calculate the probability of a specific number of events occurring within a given time interval or area using the Poisson probability formula. You can also use statistical software or tables to find probabilities for different values of lambda and event occurrences.

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